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Mirrors > Home > HSE Home > Th. List > hlimadd | Structured version Visualization version GIF version |
Description: Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlimadd.3 | ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) |
hlimadd.4 | ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) |
hlimadd.5 | ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) |
hlimadd.6 | ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) |
hlimadd.7 | ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) |
Ref | Expression |
---|---|
hlimadd | ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlimadd.3 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) | |
2 | 1 | ffvelrnda 6904 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℋ) |
3 | hlimadd.4 | . . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) | |
4 | 3 | ffvelrnda 6904 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℋ) |
5 | hvaddcl 29093 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℋ ∧ (𝐺‘𝑛) ∈ ℋ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) | |
6 | 2, 4, 5 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) |
7 | hlimadd.7 | . . . 4 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) | |
8 | 6, 7 | fmptd 6931 | . . 3 ⊢ (𝜑 → 𝐻:ℕ⟶ ℋ) |
9 | ax-hilex 29080 | . . . 4 ⊢ ℋ ∈ V | |
10 | nnex 11836 | . . . 4 ⊢ ℕ ∈ V | |
11 | 9, 10 | elmap 8552 | . . 3 ⊢ (𝐻 ∈ ( ℋ ↑m ℕ) ↔ 𝐻:ℕ⟶ ℋ) |
12 | 8, 11 | sylibr 237 | . 2 ⊢ (𝜑 → 𝐻 ∈ ( ℋ ↑m ℕ)) |
13 | nnuz 12477 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
14 | 1zzd 12208 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
15 | eqid 2737 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
16 | eqid 2737 | . . . . . 6 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
17 | 15, 16 | hhims 29253 | . . . . 5 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
18 | 15, 17 | hhxmet 29256 | . . . 4 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
19 | eqid 2737 | . . . . 5 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
20 | 19 | mopntopon 23337 | . . . 4 ⊢ ((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
21 | 18, 20 | mp1i 13 | . . 3 ⊢ (𝜑 → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
22 | hlimadd.5 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) | |
23 | 15 | hhnv 29246 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
24 | df-hba 29050 | . . . . . . 7 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
25 | 15, 23, 24, 17, 19 | h2hlm 29061 | . . . . . 6 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) |
26 | resss 5876 | . . . . . 6 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
27 | 25, 26 | eqsstri 3935 | . . . . 5 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
28 | 27 | ssbri 5098 | . . . 4 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
29 | 22, 28 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
30 | hlimadd.6 | . . . 4 ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) | |
31 | 27 | ssbri 5098 | . . . 4 ⊢ (𝐺 ⇝𝑣 𝐵 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
32 | 30, 31 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
33 | 15 | hhva 29247 | . . . . 5 ⊢ +ℎ = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
34 | 17, 19, 33 | vacn 28775 | . . . 4 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
35 | 23, 34 | mp1i 13 | . . 3 ⊢ (𝜑 → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
36 | 13, 14, 21, 21, 1, 3, 29, 32, 35, 7 | lmcn2 22546 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵)) |
37 | 25 | breqi 5059 | . . 3 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ 𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ))(𝐴 +ℎ 𝐵)) |
38 | ovex 7246 | . . . 4 ⊢ (𝐴 +ℎ 𝐵) ∈ V | |
39 | 38 | brresi 5860 | . . 3 ⊢ (𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ))(𝐴 +ℎ 𝐵) ↔ (𝐻 ∈ ( ℋ ↑m ℕ) ∧ 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵))) |
40 | 37, 39 | bitri 278 | . 2 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ (𝐻 ∈ ( ℋ ↑m ℕ) ∧ 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵))) |
41 | 12, 36, 40 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 〈cop 4547 class class class wbr 5053 ↦ cmpt 5135 ↾ cres 5553 ∘ ccom 5555 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 1c1 10730 ℕcn 11830 ∞Metcxmet 20348 MetOpencmopn 20353 TopOnctopon 21807 Cn ccn 22121 ⇝𝑡clm 22123 ×t ctx 22457 NrmCVeccnv 28665 ℋchba 29000 +ℎ cva 29001 ·ℎ csm 29002 normℎcno 29004 −ℎ cmv 29006 ⇝𝑣 chli 29008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 ax-hilex 29080 ax-hfvadd 29081 ax-hvcom 29082 ax-hvass 29083 ax-hv0cl 29084 ax-hvaddid 29085 ax-hfvmul 29086 ax-hvmulid 29087 ax-hvmulass 29088 ax-hvdistr1 29089 ax-hvdistr2 29090 ax-hvmul0 29091 ax-hfi 29160 ax-his1 29163 ax-his2 29164 ax-his3 29165 ax-his4 29166 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cn 22124 df-cnp 22125 df-lm 22126 df-tx 22459 df-hmeo 22652 df-xms 23218 df-tms 23220 df-grpo 28574 df-gid 28575 df-ginv 28576 df-gdiv 28577 df-ablo 28626 df-vc 28640 df-nv 28673 df-va 28676 df-ba 28677 df-sm 28678 df-0v 28679 df-vs 28680 df-nmcv 28681 df-ims 28682 df-hnorm 29049 df-hba 29050 df-hvsub 29052 df-hlim 29053 |
This theorem is referenced by: chscllem4 29721 |
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